Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > cffldtocusgr | Structured version Visualization version GIF version |
Description: The field of complex numbers can be made a complete simple graph with the set of pairs of complex numbers regarded as edges. This theorem demonstrates the capabilities of the current definitions for graphs applied to extensible structures. (Contributed by AV, 14-Nov-2021.) (Proof shortened by AV, 17-Nov-2021.) |
Ref | Expression |
---|---|
cffldtocusgr.p | ⊢ 𝑃 = {𝑥 ∈ 𝒫 ℂ ∣ (♯‘𝑥) = 2} |
cffldtocusgr.g | ⊢ 𝐺 = (ℂfld sSet 〈(.ef‘ndx), ( I ↾ 𝑃)〉) |
Ref | Expression |
---|---|
cffldtocusgr | ⊢ 𝐺 ∈ ComplUSGraph |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | opex 5359 | . . . . . . 7 ⊢ 〈(Base‘ndx), ℂ〉 ∈ V | |
2 | 1 | tpid1 4707 | . . . . . 6 ⊢ 〈(Base‘ndx), ℂ〉 ∈ {〈(Base‘ndx), ℂ〉, 〈(+g‘ndx), + 〉, 〈(.r‘ndx), · 〉} |
3 | 2 | orci 861 | . . . . 5 ⊢ (〈(Base‘ndx), ℂ〉 ∈ {〈(Base‘ndx), ℂ〉, 〈(+g‘ndx), + 〉, 〈(.r‘ndx), · 〉} ∨ 〈(Base‘ndx), ℂ〉 ∈ {〈(*𝑟‘ndx), ∗〉}) |
4 | elun 4128 | . . . . 5 ⊢ (〈(Base‘ndx), ℂ〉 ∈ ({〈(Base‘ndx), ℂ〉, 〈(+g‘ndx), + 〉, 〈(.r‘ndx), · 〉} ∪ {〈(*𝑟‘ndx), ∗〉}) ↔ (〈(Base‘ndx), ℂ〉 ∈ {〈(Base‘ndx), ℂ〉, 〈(+g‘ndx), + 〉, 〈(.r‘ndx), · 〉} ∨ 〈(Base‘ndx), ℂ〉 ∈ {〈(*𝑟‘ndx), ∗〉})) | |
5 | 3, 4 | mpbir 233 | . . . 4 ⊢ 〈(Base‘ndx), ℂ〉 ∈ ({〈(Base‘ndx), ℂ〉, 〈(+g‘ndx), + 〉, 〈(.r‘ndx), · 〉} ∪ {〈(*𝑟‘ndx), ∗〉}) |
6 | 5 | orci 861 | . . 3 ⊢ (〈(Base‘ndx), ℂ〉 ∈ ({〈(Base‘ndx), ℂ〉, 〈(+g‘ndx), + 〉, 〈(.r‘ndx), · 〉} ∪ {〈(*𝑟‘ndx), ∗〉}) ∨ 〈(Base‘ndx), ℂ〉 ∈ ({〈(TopSet‘ndx), (MetOpen‘(abs ∘ − ))〉, 〈(le‘ndx), ≤ 〉, 〈(dist‘ndx), (abs ∘ − )〉} ∪ {〈(UnifSet‘ndx), (metUnif‘(abs ∘ − ))〉})) |
7 | df-cnfld 20549 | . . . . 5 ⊢ ℂfld = (({〈(Base‘ndx), ℂ〉, 〈(+g‘ndx), + 〉, 〈(.r‘ndx), · 〉} ∪ {〈(*𝑟‘ndx), ∗〉}) ∪ ({〈(TopSet‘ndx), (MetOpen‘(abs ∘ − ))〉, 〈(le‘ndx), ≤ 〉, 〈(dist‘ndx), (abs ∘ − )〉} ∪ {〈(UnifSet‘ndx), (metUnif‘(abs ∘ − ))〉})) | |
8 | 7 | eleq2i 2907 | . . . 4 ⊢ (〈(Base‘ndx), ℂ〉 ∈ ℂfld ↔ 〈(Base‘ndx), ℂ〉 ∈ (({〈(Base‘ndx), ℂ〉, 〈(+g‘ndx), + 〉, 〈(.r‘ndx), · 〉} ∪ {〈(*𝑟‘ndx), ∗〉}) ∪ ({〈(TopSet‘ndx), (MetOpen‘(abs ∘ − ))〉, 〈(le‘ndx), ≤ 〉, 〈(dist‘ndx), (abs ∘ − )〉} ∪ {〈(UnifSet‘ndx), (metUnif‘(abs ∘ − ))〉}))) |
9 | elun 4128 | . . . 4 ⊢ (〈(Base‘ndx), ℂ〉 ∈ (({〈(Base‘ndx), ℂ〉, 〈(+g‘ndx), + 〉, 〈(.r‘ndx), · 〉} ∪ {〈(*𝑟‘ndx), ∗〉}) ∪ ({〈(TopSet‘ndx), (MetOpen‘(abs ∘ − ))〉, 〈(le‘ndx), ≤ 〉, 〈(dist‘ndx), (abs ∘ − )〉} ∪ {〈(UnifSet‘ndx), (metUnif‘(abs ∘ − ))〉})) ↔ (〈(Base‘ndx), ℂ〉 ∈ ({〈(Base‘ndx), ℂ〉, 〈(+g‘ndx), + 〉, 〈(.r‘ndx), · 〉} ∪ {〈(*𝑟‘ndx), ∗〉}) ∨ 〈(Base‘ndx), ℂ〉 ∈ ({〈(TopSet‘ndx), (MetOpen‘(abs ∘ − ))〉, 〈(le‘ndx), ≤ 〉, 〈(dist‘ndx), (abs ∘ − )〉} ∪ {〈(UnifSet‘ndx), (metUnif‘(abs ∘ − ))〉}))) | |
10 | 8, 9 | bitri 277 | . . 3 ⊢ (〈(Base‘ndx), ℂ〉 ∈ ℂfld ↔ (〈(Base‘ndx), ℂ〉 ∈ ({〈(Base‘ndx), ℂ〉, 〈(+g‘ndx), + 〉, 〈(.r‘ndx), · 〉} ∪ {〈(*𝑟‘ndx), ∗〉}) ∨ 〈(Base‘ndx), ℂ〉 ∈ ({〈(TopSet‘ndx), (MetOpen‘(abs ∘ − ))〉, 〈(le‘ndx), ≤ 〉, 〈(dist‘ndx), (abs ∘ − )〉} ∪ {〈(UnifSet‘ndx), (metUnif‘(abs ∘ − ))〉}))) |
11 | 6, 10 | mpbir 233 | . 2 ⊢ 〈(Base‘ndx), ℂ〉 ∈ ℂfld |
12 | cffldtocusgr.p | . . . 4 ⊢ 𝑃 = {𝑥 ∈ 𝒫 ℂ ∣ (♯‘𝑥) = 2} | |
13 | cnfldbas 20552 | . . . . . 6 ⊢ ℂ = (Base‘ℂfld) | |
14 | 13 | pweqi 4560 | . . . . 5 ⊢ 𝒫 ℂ = 𝒫 (Base‘ℂfld) |
15 | 14 | rabeqi 3485 | . . . 4 ⊢ {𝑥 ∈ 𝒫 ℂ ∣ (♯‘𝑥) = 2} = {𝑥 ∈ 𝒫 (Base‘ℂfld) ∣ (♯‘𝑥) = 2} |
16 | 12, 15 | eqtri 2847 | . . 3 ⊢ 𝑃 = {𝑥 ∈ 𝒫 (Base‘ℂfld) ∣ (♯‘𝑥) = 2} |
17 | cnfldstr 20550 | . . . 4 ⊢ ℂfld Struct 〈1, ;13〉 | |
18 | 17 | a1i 11 | . . 3 ⊢ (〈(Base‘ndx), ℂ〉 ∈ ℂfld → ℂfld Struct 〈1, ;13〉) |
19 | cffldtocusgr.g | . . 3 ⊢ 𝐺 = (ℂfld sSet 〈(.ef‘ndx), ( I ↾ 𝑃)〉) | |
20 | fvex 6686 | . . . 4 ⊢ (Base‘ndx) ∈ V | |
21 | cnex 10621 | . . . 4 ⊢ ℂ ∈ V | |
22 | 20, 21 | opeldm 5779 | . . 3 ⊢ (〈(Base‘ndx), ℂ〉 ∈ ℂfld → (Base‘ndx) ∈ dom ℂfld) |
23 | 16, 18, 19, 22 | structtocusgr 27231 | . 2 ⊢ (〈(Base‘ndx), ℂ〉 ∈ ℂfld → 𝐺 ∈ ComplUSGraph) |
24 | 11, 23 | ax-mp 5 | 1 ⊢ 𝐺 ∈ ComplUSGraph |
Colors of variables: wff setvar class |
Syntax hints: ∨ wo 843 = wceq 1536 ∈ wcel 2113 {crab 3145 ∪ cun 3937 𝒫 cpw 4542 {csn 4570 {ctp 4574 〈cop 4576 class class class wbr 5069 I cid 5462 ↾ cres 5560 ∘ ccom 5562 ‘cfv 6358 (class class class)co 7159 ℂcc 10538 1c1 10541 + caddc 10543 · cmul 10545 ≤ cle 10679 − cmin 10873 2c2 11695 3c3 11696 ;cdc 12101 ♯chash 13693 ∗ccj 14458 abscabs 14596 Struct cstr 16482 ndxcnx 16483 sSet csts 16484 Basecbs 16486 +gcplusg 16568 .rcmulr 16569 *𝑟cstv 16570 TopSetcts 16574 lecple 16575 distcds 16577 UnifSetcunif 16578 MetOpencmopn 20538 metUnifcmetu 20539 ℂfldccnfld 20548 .efcedgf 26777 ComplUSGraphccusgr 27195 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-rep 5193 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 ax-un 7464 ax-cnex 10596 ax-resscn 10597 ax-1cn 10598 ax-icn 10599 ax-addcl 10600 ax-addrcl 10601 ax-mulcl 10602 ax-mulrcl 10603 ax-mulcom 10604 ax-addass 10605 ax-mulass 10606 ax-distr 10607 ax-i2m1 10608 ax-1ne0 10609 ax-1rid 10610 ax-rnegex 10611 ax-rrecex 10612 ax-cnre 10613 ax-pre-lttri 10614 ax-pre-lttrn 10615 ax-pre-ltadd 10616 ax-pre-mulgt0 10617 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1539 df-fal 1549 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-nel 3127 df-ral 3146 df-rex 3147 df-reu 3148 df-rmo 3149 df-rab 3150 df-v 3499 df-sbc 3776 df-csb 3887 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-pss 3957 df-nul 4295 df-if 4471 df-pw 4544 df-sn 4571 df-pr 4573 df-tp 4575 df-op 4577 df-uni 4842 df-int 4880 df-iun 4924 df-br 5070 df-opab 5132 df-mpt 5150 df-tr 5176 df-id 5463 df-eprel 5468 df-po 5477 df-so 5478 df-fr 5517 df-we 5519 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 df-pred 6151 df-ord 6197 df-on 6198 df-lim 6199 df-suc 6200 df-iota 6317 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-riota 7117 df-ov 7162 df-oprab 7163 df-mpo 7164 df-om 7584 df-1st 7692 df-2nd 7693 df-wrecs 7950 df-recs 8011 df-rdg 8049 df-1o 8105 df-oadd 8109 df-er 8292 df-en 8513 df-dom 8514 df-sdom 8515 df-fin 8516 df-dju 9333 df-card 9371 df-pnf 10680 df-mnf 10681 df-xr 10682 df-ltxr 10683 df-le 10684 df-sub 10875 df-neg 10876 df-nn 11642 df-2 11703 df-3 11704 df-4 11705 df-5 11706 df-6 11707 df-7 11708 df-8 11709 df-9 11710 df-n0 11901 df-xnn0 11971 df-z 11985 df-dec 12102 df-uz 12247 df-fz 12896 df-hash 13694 df-struct 16488 df-ndx 16489 df-slot 16490 df-base 16492 df-sets 16493 df-plusg 16581 df-mulr 16582 df-starv 16583 df-tset 16587 df-ple 16588 df-ds 16590 df-unif 16591 df-cnfld 20549 df-edgf 26778 df-vtx 26786 df-iedg 26787 df-edg 26836 df-usgr 26939 df-nbgr 27118 df-uvtx 27171 df-cplgr 27196 df-cusgr 27197 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |